Parametrization of MEAM potentials for the study of metallic alloys

1. Parametrization of MEAM potentials for the study of metallic alloys G. Jomard, C. Bercegeay, S. Bernard, P.M. Anglade CEA/DIF/DPTA Laboratoire de Structure Electronique Workshop CECAM 17-19/10/2005

2. Overview • We are involved in the atomistic description of physical problems such that: • irradiation in metals and their alloys • melting of metallic alloys • solid state phase transformations • Molecular dynamics simulations had proven to be a powerful technique to treat these kind of problems. • But they require a relevant atomic interaction model. • Our physical systems are composed of either f-elements (Pu) or BCC transition metals (Ta,W) •  the MEAM formalism of M. Baskes is known to correctly describe these metals • for which there is a strong covalent part in the atomic bonding. • Among the various MEAM formulations we have chosen the 2nd NN one which has been proven to • greatly improve physical description of BCC metals compare to the 1st NN one. • The parametrization of our semi-empirical potentials is based on DFT calculations. • For the ab initio part of this work we used the VASP code (PAW pseudopotentials, GGA corrected • exchange correlation with PBE functional). Workshop CECAM 17-19/10/2005

3. MEAM formalism for one component materials The total energy of a system of N particles is approximated as: Workshop CECAM 17-19/10/2005

4. MEAM formalism for one component materials Background electron density on site i i j-1 j 3 rij i 1 2 electrons gas The total energy of a system of N particles is approximated as: Embedding energy N-body term based on the Pauling’s logarithmic relationship between bond length and number of bonds Workshop CECAM 17-19/10/2005

5. MEAM formalism for one component materials is composed of : Background electron density on site i • Spherically symmetric partial electron density • Angular contributions of symmetry p,d and f. The total energy of a system of N particles is approximated as: Embedding energy N-body term Workshop CECAM 17-19/10/2005

6. MEAM formalism for one component materials EOS at T=0K (DFT) The total energy of a system of N particles is approximated as: Pair interaction between atom i and j Assuming that for a given reference structure, the energy per atom follows the URBE of Rose: The pair potential between 2 atoms separated by r is: 1st NN coordinence Workshop CECAM 17-19/10/2005

7. Cutting the interactions in MEAM Usually we restrict the interactions to 1st NN by applying angular screening between atoms i and j. The screening of the interaction ij by atom k is limited by the introduction of 2 parameters Cmin and Cmax. Cmax=2.8 and Cmin=2.0 ensure that for FCC structure 1st NN are completely unscreened and interactions are still 1stNN for other structures like BCC, HCP,… Note that reducing Cmin allows to take into account 2NN interactions This has be proven to improve the physical description of BCC materials for which d2ndNN is only 15% grater than d1stNN. In this case, the pair potential is obtained from a converging serie: In addition, for computational convenience, a radial cut-off is also applied to and . Workshop CECAM 17-19/10/2005

8. Parametrization of 2ndNN MEAM potentials for Ta and W As seen previously, there are 14 independent parameters in the 2nd NN MEAM formalism. The parametrization procedure is: • Adjusting the URBE of Rose on the DFT calculated EOS of BCC Ta and W •  E0, r0 and • Adjusting the , , A, Cmin and Cmax on physical properties such as • the elastic constants (C11, C12 and C44), , and . • We use a code which searches a set of parameters minimizing a sum of • qualities defined by: DFT calculated value relative error allowed Note that for this study, we have partially used the potentials published by B.J. Lee and al. [PRB 64 (2000) 184102] since our purpose was to parametrize the cross potential Workshop CECAM 17-19/10/2005

9. Parametrization of a 2ndNN MEAM potential for the Ta-W interactions The first step of the parametrization procedure is to choose a reference structure in order to obtain an analytical formulation for For convenience, a perfectly ordered binary intermetallic compound where an atom of type A has only B-type 1st NN and A-type 2ndNN, may be a good choice B1 (NaCl structure) or B2 (CsCl structure) We chose the B2 structure for which we can write: Ta W Workshop CECAM 17-19/10/2005

10. Parametrization of a 2ndNN MEAM potential for the Ta-W interactions URBE of Rose - Sikj calculated with and if ai=aj. - Sikj calculated with and if aiaj. • Sequential procedure : • parametrization of ETaW • determination of STa and SW within B2 compound  STaWTa and SWTaW • In a first approximation we have tried: pre-determined one-component functions 4 parameters determined by least squares minimisation within this methodthe screening do not depend on the type of atom k! Workshop CECAM 17-19/10/2005

11. Parametrization of a 2ndNN MEAM potential for the Ta-W interactions We adjust the parameters of the Rose’s equation on the EOS of TaW B2 obtained by first principles electronic structure calculations. For the calculation of the screening in TaW compounds, we use - Sikj calculated with and if ai=aj. - Sikj calculated with and if aiaj. 1st attempt: the crude one. Workshop CECAM 17-19/10/2005

12. Parametrization of a 2ndNN MEAM potential for the Ta-W interactions 1st attempt: the crude one. The agreement with DFT results is not so good! But there are many crude assumptions in this parametrization of the cross potential Workshop CECAM 17-19/10/2005

13. Parametrization of a 2ndNN MEAM potential for the Ta-W interactions i • Assumption 1: In the construction of the background density on a site i, the weights given to partial densities induced by the neighbors is only dependent of the type of atom i.  we should introduce weights depending on the local atomic arrangement around site i with Ta W depends on the types of the neighbors of atom i depends only of type of atom i R we tried: 6 new parameters no new parameters, only an average on local atomic surrounding Workshop CECAM 17-19/10/2005

14. Parametrization of a 2ndNN MEAM potential for the Ta-W interactions formation energies of various compounds Ta15W, Ta3W DO3, Ta3W SQS, TaW B2, TaW SQS, TaW3 DO3, TaW3 SQS and TaW15. • Assumption 2: The screening parameters depend only on the type of the two atoms which interact and note on the type of the screening atom.  introduction of more parameters : in order to calculate Sikj this implies to increase the set of physical properties entering the parametrization Workshop CECAM 17-19/10/2005

15. Parametrization of a 2ndNN MEAM potential for the Ta-W interactions We adjust the parameters of the Rose’s equation on the EOS of TaW B2 obtained by first principles electronic structure calculations. We adjust all the other parameters : , the and the on the formation energies of a set of ideal and relaxed TaW compounds calculated with VASP : Ta15W, Ta3W DO3, Ta3W SQS, TaW B2, TaW SQS, TaW3 DO3, TaW3 SQS and TaW15. 2nd attempt: The overall agreement with DFT results is much more better. The worse agreement is found for the 94at% of W composition. But many compounds scratch with temperature! Workshop CECAM 17-19/10/2005

16. Parametrization of a 2ndNN MEAM potential for the Ta-W interactions 3rd attempt: We adjust the parameters of the Rose’s equation on the EOS of TaW B2 obtained by first principles electronic structure calculations. We adjust all the other parameters : and the on the formation energies of a set of ideal and relaxed TaW compounds calculated with VASP : Ta15W, Ta3W DO3, Ta3W SQS, TaW B2, TaW SQS, TaW3 DO3, TaW3 SQS and TaW15. We use the mean weights for partial densities. Workshop CECAM 17-19/10/2005

17. Parametrization of a 2ndNN MEAM potential for the Ta-W interactions The agreement between DFT data and the ones obtained with the fitted potential is excellent. All the structures are stable with temperature. Is this potential transferable to other crystallographic structures? Workshop CECAM 17-19/10/2005

18. Transferability of the potential? 0 -0.05 -0.10 Formation Energy (eV/at) -0.15 -0.20 -0.25 Ta composition (at%) Blum and Zunger PRB 72 (2005) 020104 Exp @ 1200K Workshop CECAM 17-19/10/2005